HINT: <no title>
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Use the segment CD⎯⎯⎯⎯⎯⎯⎯⎯
to get two right-angled triangles. Then you can use trigonometric
ratios or the theorem of Pythagoras to work out the answer to the
question.
STEP: Draw a line to create right-angled triangles
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The first thing to do is draw an extra line
across the triangle so that we make two right-angled triangles in the
figure. We do this because we can use the trigonometric ratios and the
theorem of Pythagoras for right-angled triangles.
The line segment DC⎯⎯⎯⎯⎯⎯⎯⎯
in the figure is the line we want: it will create two separate
right-angled triangles! The two right-angled triangles that we get look
like this:
STEP: Use trigonometry to find useful information
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In ΔADC
(the light blue one) we know one of the non-right angles and one of the
sides. Hence we can use the trigonometric ratios in that triangle
because it is a right-angled triangle.
With the information given, we can find both segments AD⎯⎯⎯⎯⎯⎯⎯⎯ and CD⎯⎯⎯⎯⎯⎯⎯⎯. Remember that we want to get the measure of B̂ , and for that we need both of these lengths. Start by calculating the length of AD⎯⎯⎯⎯⎯⎯⎯⎯, which will allow us to find DB⎯⎯⎯⎯⎯⎯⎯⎯ (because we know that AB⎯⎯⎯⎯⎯⎯⎯=9,7). This calculation involves the hypotenuse and the side adjacent to  , so use the cosine ratio.
cosθcos(58,2°)(8,2)cos58,2°(8,2)(0,5269...)4,3210...SinceAD⎯⎯⎯⎯⎯⎯⎯⎯+DB⎯⎯⎯⎯⎯⎯⎯⎯=AB⎯⎯⎯⎯⎯⎯⎯:=adjacenthypotenuse=AD⎯⎯⎯⎯⎯⎯⎯⎯8,2=AD⎯⎯⎯⎯⎯⎯⎯⎯=AD⎯⎯⎯⎯⎯⎯⎯⎯=AD⎯⎯⎯⎯⎯⎯⎯⎯DB⎯⎯⎯⎯⎯⎯⎯⎯=9,7−4,3210...DB⎯⎯⎯⎯⎯⎯⎯⎯=5,3789...
Great: that gets us the value for side DB⎯⎯⎯⎯⎯⎯⎯⎯. Now we need to find the length of side CD⎯⎯⎯⎯⎯⎯⎯⎯. For that we will use the sine ratio (you can also do this calculation with the theorem of Pythagoras, but here we will do it with trigonometry).
sinθsin(58,2°)(8,2)sin58,2°(8,2)(0,8498...)6,9691...=oppositehypotenuse=CD⎯⎯⎯⎯⎯⎯⎯⎯8,2=CD⎯⎯⎯⎯⎯⎯⎯⎯=CD⎯⎯⎯⎯⎯⎯⎯⎯=CD⎯⎯⎯⎯⎯⎯⎯⎯
STEP: Calculate the final answer
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Now we can finally work out the angle that we want. CD⎯⎯⎯⎯⎯⎯⎯⎯=6,9691... and DB⎯⎯⎯⎯⎯⎯⎯⎯=5,3789... are the opposite and adjacent sides for angle B̂ , respectively, so this is a tangent ratio situation.
tanθtanB̂ B̂ =oppositeadjacent=6,9691...5,3789...=tan−1(6,9691...5,3789...)=52,3380...≈52,3°
Remember that the instructions say to round the
answer to the first decimal place, as shown in the last step above.
The final answer is: B̂ =52,3°.
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